Question

The number of points with integral coordinates that are interior to the circle $x^2+y^2=16$ is

• A. 43
• B. 49
• C. 45
• D. 51

Answer 1

The correct option is C 45

We are asked to find the integral co-ordinates for (x, y) interior to the circle $x^2+y^2=16$ or satisfying $x^2+y^2$ < 16
Well, to answer this question we'll see all the integral coordiantes and check it by putting the values in the equation of the circle.
Co-ordinates will be -
(0,0), (0,1),(0,2), (0,3)
(1,0), (1,1),(1,2), (1,3)
(2,0), (2,1),(2,2), (2,3)
(3,0) (3,1) (3,2)
Out of these 15 points 4 points lie on the x- axis including origin and 3 points lie on the Y-axis excluding origin.
Therefore after excluding these points we have 8 points in the 1st quadrant which are inside the circle and not lying on any axis.
We can directly say the points inside all the four quadrants will be nothing but = 8*4 = 32
Now we have to account the points which are lying on axes.
On X- axis we have 7 points including origin.
On Y - axis we have 6 points excluding origin.
So total points will be = 32 + 7 + 6 = 45